Principles and demonstrations of quantum information processing by NMR spectroscopy

This paper surveys our recent research on quantum information processing by nuclear magnetic resonance (NMR) spectroscopy. We begin with a geometric i ntroduction to the NMR of an ensemble of indistinguishable spins, and then show how this geometric interpretation is contained within an algebra of mu ltispin product operators. This algebra is used throughout the rest of the paper to demonstrate that it provides a facile framework within which to st udy quantum information processing more generally. The implementation of qu antum algorithms by NMR depends upon the availability of special kinds of m ixed states, called pseudo-pure states, and we consider a number of differe nt methods for preparing these states, along with analyses of how they scal e with the number of spins. The quantum-mechanical nature of processes invo lving such macroscopic pseudo-pure states also is a matter of debate, and i n order to discuss this issue in concrete terms we present the results of N MR experiments which constitute a macroscopic analogue Hardy's paradox. Fin ally, a detailed product operator description is given of recent NMR experi ments which demonstrate a three-bit quantum error correcting code, using fi eld gradients to implement a precisely-known decoherence model.